DECOMPOSITION OF COMPLETE GRAPHS AND COMPLETE BIPARTITE GRAPHS INTO COPIES OF P sub n sub sup 3 sup or S sub 2 sub P sub n sub sup 3 sup AND HARMONIOUS LABELING OF K sub 2 sub + P sub n sub | Selvaraju | Journal of the Indonesian Mathematical
J. Indones. Math. Soc.
Special Edition (2011), pp. 109–122.
DECOMPOSITION OF COMPLETE GRAPHS AND
COMPLETE BIPARTITE GRAPHS INTO COPIES OF Pn3
OR S2 (Pn3 ) AND HARMONIOUS LABELING OF K2 + Pn
P. Selvaraju1 and G. Sethuraman2
1
2
Department of Mathematics, VEL TECH, Avadi, Chennai-600062, India,
[email protected]
Department of Mathematics, Anna University, Chennai-600025, Chennai,
India, [email protected]
Abstract. In this paper, the graphs Pn3 and S2 (Pn3 ) are shown to admit an αvaluation, where Pn3 is the graph obtained from the path Pn by joining all the pairs
of vertices u, v of Pn with d(u, v) = 3 and S2 (Pn3 ) is the graph obtained from Pn3 by
merging the centre of the star Sn1 and that of the star Sn2 respectively at the two
unique 2-degree vertex of Pn3 (the origin and terminus of the path Pn contained in
Pn3 ). It follows from the significant theorems due to Rosa [1967] and EI-Zanati and
Vanden Eynden [1996] that the complete graphs K2cq+1 or the complete bipartite
graphs Kmq,nq can be cyclically decomposed into the copies of Pn3 or copies of
S2 (Pn3 ), where c, m, n are arbitrary positive integer and q denotes either |E(Pn3 )|
or |E(S2 (Pn3 ))|. Further, it is shown that join of complete graph K2 and path Pn ,
denoted K2 + Pn , for n ≥ 1 is harmonious graph.
Key words: α-labeling, harmonious labeling, Pn3 graphs, join, path.
2000 Mathematics Subject Classification: 05C78.
Received: 09-08-2011, revised: 09-09-2011, accepted: 04-12-2012.
Special Edition (2011), pp. 109–122.
DECOMPOSITION OF COMPLETE GRAPHS AND
COMPLETE BIPARTITE GRAPHS INTO COPIES OF Pn3
OR S2 (Pn3 ) AND HARMONIOUS LABELING OF K2 + Pn
P. Selvaraju1 and G. Sethuraman2
1
2
Department of Mathematics, VEL TECH, Avadi, Chennai-600062, India,
[email protected]
Department of Mathematics, Anna University, Chennai-600025, Chennai,
India, [email protected]
Abstract. In this paper, the graphs Pn3 and S2 (Pn3 ) are shown to admit an αvaluation, where Pn3 is the graph obtained from the path Pn by joining all the pairs
of vertices u, v of Pn with d(u, v) = 3 and S2 (Pn3 ) is the graph obtained from Pn3 by
merging the centre of the star Sn1 and that of the star Sn2 respectively at the two
unique 2-degree vertex of Pn3 (the origin and terminus of the path Pn contained in
Pn3 ). It follows from the significant theorems due to Rosa [1967] and EI-Zanati and
Vanden Eynden [1996] that the complete graphs K2cq+1 or the complete bipartite
graphs Kmq,nq can be cyclically decomposed into the copies of Pn3 or copies of
S2 (Pn3 ), where c, m, n are arbitrary positive integer and q denotes either |E(Pn3 )|
or |E(S2 (Pn3 ))|. Further, it is shown that join of complete graph K2 and path Pn ,
denoted K2 + Pn , for n ≥ 1 is harmonious graph.
Key words: α-labeling, harmonious labeling, Pn3 graphs, join, path.
2000 Mathematics Subject Classification: 05C78.
Received: 09-08-2011, revised: 09-09-2011, accepted: 04-12-2012.